XIV
PREFACE
to an approach close to the original one of Bender and Aschbacher, for the sake of
efficiency.
Chapter 3 is devoted to five "p-component pre-uniqueness theorems", Theorems
PU1-PU5. The common theme is a maximal subgroup M which has a p-component
K such that the centralizer W of KjOv (K) has p-rank at least 2 and such that
CG(V) M for every non-identity p-element of W. This type of situation will
arise typically when K is maximal in some ordering on the set of p-components of
centralizers of p-elements of G, and the p'-cores
O
P
/(GG(X)) , X
G W, have already
been assembled, for example by the signalizer functor method or by an assumption
that they are trivial. The generic conclusion is that M is a strongly p-embedded
subgroup of G, which yields an immediate contradiction if p = 2. When p is odd
and the DC-proper simple group G has even type, it will later be shown that again G
does not exist. Thus the eventual import of Chapter 3 will be to establish in general
that maximal p-components of centralizers of p-elements have centralizers of p-rank
1. (Of course there are counterexamples to this statement, for example when p = 2
and G is an alternating group.) Historically results of this type were established
first for p = 2 by Powell and Thwaites, whose ideas are incorporated into Section
17 of Chapter 3. Shortly thereafter, Aschbacher proved his Component Theorem, a
more definitive version for p = 2. Robert Gilman gave a somewhat different proof
of Aschbacher's result. Many of the ideas of Aschbacher and Gilman also appear
here, but some of their delicate analysis is replaced by a detailed consideration of
DC-groups. The results proved here are new in the case that p is odd. We remark
that in the proof of the first major pre-uniqueness theorem, Theorem PUi, the
case p = 2 is handled fairly quickly thanks to Aschbacher's criterion for a strongly
embedded subgroup (Theorem ZD). Thus in Sections 7-15 of Chapter 3 the prime
p is odd.
The structure and embedding of p-component uniqueness subgroups when p = 2
and K has 2-rank one is somewhat exceptional. In particular the main assertion of
Theorem PU4, that terminal components are standard, is not valid in this situa-
tion. In other words, K could commute elementwise with a conjugate. Historically
Aschbacher and Richard Foote were able to show that there could be only one
such conjugate, and we prove a similar result in Theorem PU5. Thanks are due to
Professor Foote, who suggested years ago that such a result ought to be simple to
prove.
As noted above, with the exception of Theorem ZD and its corollaries Theorems
SE and SZ, all the results are proved for a DC-proper simple group G. The proof
thus can and does rely heavily on the theory of almost simple DC-groups extablished
in pU]. Those DC-group properties essential for Chapters 2 and 3 are collected
in Chapter 4 of this volume and either extend or follow directly from the theory
presented in our preceding volume [IA] . A much briefer Chapter 1 similarly extends
our second volume [IQ] with some "general" (as opposed to DC-group theoretic)
results pertinent to our task. Notable here is some theory of permutation groups
underlying the proof of Theorem ZD.
In references (even within this volume), we shall specify the four chapters of
this volume as Hi, II2, II3 and II4, respectively.
We are grateful to Michael O'Nan for his assistance with the proof of Theorem
3.2 of Chapter 1; to Sergey Shpectorov for his enthusiastic support and constructive
comments during the preparation of Chapter 2; to Michael Aschbacher and Hel-
mut Bender for their support and helpful comments and suggestions; and to Inna
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